## The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains.(English)Zbl 1185.65210

Authors’ abstract: We analyze a generalization of the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. The method was originally introduced by W. E and B. Engquist [Commun. Math. Sci. 1, No. 1, 87–132 (2003; Zbl 1093.35012)] for homogenization problems in fixed domains. It is based on a standard finite element approach on the macroscale, where the stiffness matrix is computed by solving local cell problems on the microscale. A posteriori error estimates are derived in $$L^{2}(\Omega)$$ by reformulating the problem into a discrete two-scale formulation [see also M. Ohlberger, Multiscale Model. Simul. 4, No. 1, 88–114 (2005; Zbl 1090.65128)] and using duality methods afterwards. Numerical experiments are given in order to numerically evaluate the efficiency of the error estimate.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure

### Citations:

Zbl 1093.35012; Zbl 1090.65128
Full Text:

### References:

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